We show that the Kazhdan-Lusztig basis elements Cw of the Hecke algebra of the symmetric group, when w ∈ Sn corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form Ti + fj(v), where fj are rational functions.

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness Ž of the infinite cluster is equivalent to nondecay of connectivity a.k.a. . long-range order . We then derive applications concerning uniqueness in Kazhdan groups and in wreath products ...

We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its N -region. In type An this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells.

The goal of this paper is to give a simple proof of Deligne’s conjecture on the Lefschetz trace formula (proven by Fujiwara) and to generalize it to the situation appearing in the forthcoming joint paper with D. Kazhdan. Our proof holds in the realm of ordinary algebraic geometry and does not use rigid geometry.

Abstract We prove that an integral block of the category $\mathcal O$ periplectic Lie superalgebra contains a non-semisimple Jantzen middle if and only it simple module atypical highest weight. As consequence, every does not admit Kazhdan–Lusztig theory in sense Cline, Parshall, Scott.